Discrete Mathematics and Combinatorics Commons^™

(R2139) Italian Domination Number For Some Classes Of Trees, Amritha Prakash, Ragukumar Pandurangan

Applications and Applied Mathematics: An International Journal (AAM)

For a graph G with vertex set V , an Italian dominating function is a function f from V to {0, 1, 2} which has the property that for every vertex which is assigned 0, it must either adjacent to a vertex assigned 2 under f or adjacent to at least two vertices assigned 1 under f. The weight of an Italian dominating function is the sum of all weights assigned to the vertices. The minimum weight of an Italian dominating function f is the Italian domination number. Finding a graph’s Italian domination number is a well-known NP-Complete problem. Even …

On Variations Of Isolation In Graphs, Geoffrey Boyer

All Dissertations

In 2015, Caro and Hansberg introduced a wonderful and natural generalization of the well-studied parameter of domination in graphs. For a graph $G$ and a family of graphs $\FF$, they define $S$ to be an $\FF$-isolating set if $G-N[S]$ contains no member of $\FF$ as a subgraph. The case where $\FF=\{K_1\}$ coincides with domination. The case where $\FF=\{K_2\}$ is now simply referred to as an isolating set. We strengthen known results about the isolation number of a graph, and explore variations of the parameter including the independent and total versions.

In particular for connected graphs of order $n$, a bound …

Asteroidal Sets And Dominating Targets In Graphs, Oleksiy Al-Saadi

School of Computing: Dissertations, Theses, and Student Research

The focus of this PhD thesis is on various distance and domination properties in graphs. In particular, we prove strong results about the interactions between asteroidal sets and dominating targets. Our results add to or extend a plethora of results on these properties within the literature. We define the class of strict dominating pair graphs and show structural and algorithmic properties of this class. Notably, we prove that such graphs have diameter 3, 4, or contain an asteroidal quadruple. Then, we design an algorithm to to efficiently recognize chordal hereditary dominating pair graphs. We provide new results that describe the …

Domination In Graphs And The Removal Of A Matching, Geoffrey Boyer

All Theses

We consider how the domination number of an undirected graph changes on the removal of a maximal matching. It is straightforward that there are graphs where no matching removal increases the domination number, and where some matching removal doubles the domination number. We show that in a nontrivial tree there is always a matching removal that increases the domination number; and if a graph has domination number at least $2$ there is always a maximal matching removal that does not double the domination number. We show that these results are sharp and discuss related questions.

Δ-Small Intersection Graphs Of Modules, Ahmed H. Alwan

Al-Bahir

Let R be a commutative ring with unit and M be a unitary left R-module. The δ-small intersection graph of non-trivial submodules of , denoted by , is an undirected simple graph whose vertices are the non-trivial submodules of , and two vertices are adjacent if and only if their intersection is a -small submodule of . In this article, we study the interplay between the algebraic properties of , and the graph properties of such as connectivity, completeness and planarity. Moreover, we determine the exact values of the diameter and girth of , as well as give a formula …

Irregular Domination In Graphs, Caryn Mays

Dissertations

Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters …

Investigations In The Semi-Strong Product Of Graphs And Bootstrap Percolation, Kevin J. Mccall

Theses and Dissertations

The semi-strong product of graphs G and H is a way of forming a new graph from the graphs G and H. The vertex set of the semi-strong product is the Cartesian product of the vertex sets of G and H, V(G) x V(H). The edges of the semi-strong product are determined as follows: (g₁,h₁)(g₂,h₂) is an edge of the product whenever g₁g₂ is an edge of G and h₁h₂ is an edge of H or g₁ = g₂ and h₁h_{2 …}

An Improvement In The Two-Packing Bound Related To Vizing's Conjecture, Kimber Wolff

Theory & Applications of Graphs

Vizing's conjecture states that the domination number of the Cartesian product of graphs is at least the product of the domination numbers of the two factor graphs. In this note we improve the recent bound of Breŝar by applying a technique of Zerbib to show that for any graphs G and H, γ(G x H)≥ γ (G) 2/3(γ(H)-ρ(H)+1), where γ is the domination number, ρ is the 2-packing number, and x is the Cartesian product.

Asymptotically Optimal Bounds For (𝑡,2) Broadcast Domination On Finite Grids, Timothy W. Randolph

Rose-Hulman Undergraduate Mathematics Journal

Let G = (V,E) be a graph and t,r be positive integers. The signal that a tower vertex T of signal strength t supplies to a vertex v is defined as sig(T, v) = max(t − dist(T,v),0), where dist(T,v) denotes the distance between the vertices v and T. In 2015 Blessing, Insko, Johnson, and Mauretour defined a (t, r) broadcast dominating set, or simply a (t, r) broadcast, on G as a set T ⊆ V such that the sum of all signal received at each vertex v ∈ V from the set of towers T …

Reducing The Maximum Degree Of A Graph By Deleting Vertices: The Extremal Cases, Peter Borg, Kurt Fenech

Theory & Applications of Graphs

Let λ(G) denote the smallest number of vertices that can be removed from a non-empty graph G so that the resulting graph has a smaller maximum degree. In a recent paper, we proved that if n is the number of vertices of G,k is the maximum degree of G, and t is the number of vertices of degree k, then λ: (G) ≤ n+(k-1)t}{2k}. We also showed that λ (G) ≤ \frac{n}{k+1} if G is a tree. In this paper, we provide a new proof of the first bound and use it to determine the graphs that …

Automated Conjecturing Approach For Benzenoids, David Muncy

Theses and Dissertations

Benzenoids are graphs representing the carbon structure of molecules, defined by a closed path in the hexagonal lattice. These compounds are of interest to chemists studying existing and potential carbon structures. The goal of this study is to conjecture and prove relations between graph theoretic properties among benzenoids. First, we generate conjectures on upper bounds for the domination number in benzenoids using invariant-defined functions. This work is an extension of the ideas to be presented in a forthcoming paper. Next, we generate conjectures using property-defined functions. As the title indicates, the conjectures we prove are not thought of on our …

Distance-2 Domatic Numbers Of Graphs, Derek Kiser

Electronic Theses and Dissertations

The distance d(u, v) between two vertices u and v in a graph G equals the length of a shortest path from u to v. A set S of vertices is called a distance-2 dominating set if every vertex in V \S is within distance-2 of at least one vertex in S. The distance-2 domatic number is the maximum number of sets in a partition of the vertices of G into distance-2 dominating sets. We give bounds on the distance-2 domatic number of a graph and determine the distance-2 domatic number of selected classes of graphs.

Domination Numbers Of Semi-Strong Products Of Graphs, Stephen R. Cheney

Theses and Dissertations

This thesis examines the domination number of the semi-strong product of two graphs G and H where both G and H are simple and connected graphs. The product has an edge set that is the union of the edge set of the direct product of G and H together with the cardinality of V(H), copies of G. Unlike the other more common products (Cartesian, direct and strong), the semi-strong product is neither commutative nor associative.

The semi-strong product is not supermultiplicative, so it does not satisfy a Vizing like conjecture. It is also not submultiplicative so it shares these two …

The K-Dominating Graph, Ruth Haas, Karen Seyffarth

Mathematics Sciences: Faculty Publications

Abstract. Given a graph G, the k-dominating graph of G, D_k(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D_k(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D_k(G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of …

Locating-Domination In Complementary Prisms., Kristin Renee Stone Holmes

Electronic Theses and Dissertations

Let G = (V (G), E(G)) be a graph and G̅ be the complement of G. The complementary prism of G, denoted GG̅, is the graph formed from the disjoint union of G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. A set D ⊆ V (G) is a locating-dominating set of G if for every u ∈ V (G)D, its neighborhood N(u)⋂D is nonempty and distinct from N( …

Double Domination Of Complementary Prisms., Lamont D. Vaughan

Electronic Theses and Dissertations

The complementary prism of a graph G is obtained from a copy of G and its complement G̅ by adding a perfect matching between the corresponding vertices of G and G̅. For any graph G, a set D ⊆ V (G) is a double dominating set (DDS) if that set dominates every vertex of G twice. The double domination number, denoted γ_×2(G), is the cardinality of a minimum double dominating set of G. We have proven results on graphs of small order, specific families and lower bounds on γ_{×2 …}

Alliance Partitions In Graphs., Jason Lachniet

Electronic Theses and Dissertations

For a graph G=(V,E), a nonempty subset S contained in V is called a defensive alliance if for each v in S, there are at least as many vertices from the closed neighborhood of v in S as in V-S. If there are strictly more vertices from the closed neighborhood of v in S as in V-S, then S is a strong defensive alliance. A (strong) defensive alliance is called global if it is also a dominating set of G. The alliance partition number (respectively, strong …

Double Domination Edge Critical Graphs., Derrick Wayne Thacker

Electronic Theses and Dissertations

In a graph G=(V,E), a subset S ⊆ V is a double dominating set if every vertex in V is dominated at least twice. The minimum cardinality of a double dominating set of G is the double domination number. A graph G is double domination edge critical if for any edge uv ∈ E(G̅), the double domination number of G+uv is less than the double domination number of G. We investigate properties of double domination edge critical graphs. In particular, we characterize the double domination edge critical trees and …